2019/07/17 by Hanumanthu, Krishna, Harbourne, Brian
#14C17 #14J26 #14N20 #52C30 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.07712
We study supersolvable line arrangements in \mathbb P2 over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil) complex line arrangement cannot have more than 4 modular points, and if all of the crossing points of a complex line arrangement have multiplicity 3 or 4, then the arrangement must have 0 modular points (i.e., it cannot be supersolvable). This provides at least a little evidence for our conjecture that every nontrivial complex supersolvable line arrangement has at least one point of multiplicity 2, which in turn is a step toward the much stronger conjecture of Anzis and Tohǎneanu that every nontrivial complex supersolvable line arrangement with s lines has at least s/2 points of multiplicity 2.